The solution is simply that there is no such barber (and neither could there be).
It's not really a paradox because the solution is so simple (as Russell himself notes in, I think, *Philosophy of Logical Atomism*--it's not actually his paradox, but was suggested to him as a paradox and he disagreed).
The real Russellian paradoxes begin with the paradox of the class of classes that are not members of themselves. Is this class a member of itself? If yes, then no; if no, then yes. Contradiction.
(How about the property of being a property that doesn't characterize itself?
There are many more.)
Like the Barber, the simple class paradox can be superficially 'solved' by saying there's no such class (as opposed to Barber). But now we have a real paradox because it's not at all clear why there should be no such class.
